Problem: Let $g(x)=\dfrac{\ln(x)}{x}$. What is the absolute maximum value of $g$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $-e$ (Choice B) B $\dfrac{1}{e}$ (Choice C) C $-\dfrac{1}{e^2}$ (Choice D) D $g$ has no maximum value
Explanation: Let's first find the relative extremum points of $g$, and then consider them along with the function's end behavior in both directions. Note that the domain of $g$ is all real numbers such that $x>0$. We start with finding the critical points of $g$. The derivative of $g$ is $g'(x)=\dfrac{1-\ln(x)}{x^2}$. $g'(x)=0$ for $x=e$. $g'$ is defined for all real numbers in the domain of $g$. Therefore, our only critical point is $x=e$. Our critical point divides the function's domain (which is all real numbers such that $x>0$ ) into two intervals: $0$ $1$ $2$ $3$ $4$ $(0, e)$ $(e, \infty)$ $e$ Let's evaluate $g'$ at each interval to see if it's positive or negative on that interval. Interval $x$ -value $g'(x)$ Verdict $(0,e)$ $x=1$ $g'(1)=1>0$ $g$ is increasing $\nearrow$ $(e,\infty)$ $x=4$ $g'(4)=\dfrac{1-\ln(4)}{16}<0$ $g$ is decreasing $\searrow$ Let's imagine ourselves walking on the graph of $g$, starting at the left end of the domain $($ which is $0$ $)$ and going all the way to the right (until $+\infty$ ). According to the table, we will start by going up until we reach $x=e$. Then, we will be forever going down. Therefore, $g$ must obtain its absolute maximum value at $x=e$. We are asked to find that maximum value, which is $g\left(e\right)=\dfrac{1}{e}$. In conclusion, the absolute maximum value of $g$ is $\dfrac{1}{e}$.